prompt stringlengths 41 2k | solution stringlengths 1 22 | data_source stringclasses 1
value | source_prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict | content listlengths 1 1 |
|---|---|---|---|---|---|---|---|
Let $O$ and $H$ be the circumcenter and orthocenter of triangle $ABC$, respectively. Let $a$, $b$, and $c$ denote the side lengths, and let $R$ denote the circumradius. Find $OH^2$ if $R = 7$ and $a^2 + b^2 + c^2 = 29$. | 412 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $O$ and $H$ be the circumcenter and orthocenter of triangle $ABC$, respectively. Let $a$, $b$, and $c$ denote the... | MATH | {
"ground_truth": "412",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b9b37c9a-b1d6-48c5-9168-6cb89dabfcd1"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $O$ and $H$ be the circumcenter and orthocenter of triangle $ABC$, respectively. Let $a$, $b$, and $c$ denote the... |
Let $A_1A_2A_3 \ldots A_{21}$ be a 21-sided regular polygon inscribed in a circle with center $O$. How many triangles $A_iA_jA_k$, $1 \leq i < j < k \leq 21$, contain the center point $O$ in their interior? | 385 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $A_1A_2A_3 \\ldots A_{21}$ be a 21-sided regular polygon inscribed in a circle with center $O$. How many triangles... | MATH | {
"ground_truth": "385",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "dd315c09-6465-4836-ae8d-c427a374f1f4"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $A_1A_2A_3 \\ldots A_{21}$ be a 21-sided regular polygon inscribed in a circle with center $O$. How many triangles... |
Suppose that the angles of $\triangle ABC$ satisfy $\cos(3A)+\cos(3B)+\cos(3C)=1.$ Two sides of the triangle have lengths $10$ and $13$. There is a positive integer $m$ so that the maximum possible length for the remaining side of $\triangle ABC$ is $\sqrt{m}.$ Find $m.$ | 399 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose that the angles of $\\triangle ABC$ satisfy $\\cos(3A)+\\cos(3B)+\\cos(3C)=1.$ Two sides of the triangle have ... | MATH | {
"ground_truth": "399",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "aa9b5424-63a5-4400-94bf-188e2b420216"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose that the angles of $\\triangle ABC$ satisfy $\\cos(3A)+\\cos(3B)+\\cos(3C)=1.$ Two sides of the triangle have ... |
Two points $A(x_1, y_1)$ and $B(x_2, y_2)$ are chosen on the graph of $f(x) = \ln x$, with $0 < x_1 < x_2$. The points $C$ and $D$ trisect $\overline{AB}$, with $AC < CB$. Through $C$ a horizontal line is drawn to cut the curve at $E(x_3, y_3)$. Find $x_3$ if $x_1 = 1$ and $x_2 = 1000$. | 10 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTwo points $A(x_1, y_1)$ and $B(x_2, y_2)$ are chosen on the graph of $f(x) = \\ln x$, with $0 < x_1 < x_2$. The point... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "37f3020f-fd4c-4bba-aa34-9fe13a91ffda"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTwo points $A(x_1, y_1)$ and $B(x_2, y_2)$ are chosen on the graph of $f(x) = \\ln x$, with $0 < x_1 < x_2$. The point... |
A set of positive integers is said to be *pilak* if it can be partitioned into two disjoint subsets $F$ and $T$, each with at least two elements, such that the elements of $F$ are consecutive Fibonacci numbers, and the elements of $T$ are consecutive triangular numbers.
Find all positive integers $n$ such that the se... | 30 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA set of positive integers is said to be *pilak* if it can be partitioned into two disjoint subsets $F$ and $T$, each ... | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4c90e78c-bc5d-4cd3-a831-8e76f4a1d966"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA set of positive integers is said to be *pilak* if it can be partitioned into two disjoint subsets $F$ and $T$, each ... |
Find all prime numbers that do not have a multiple ending in $2015$. | 2 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all prime numbers that do not have a multiple ending in $2015$.\n\nRemember to put your answer on its own line af... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "500daa08-be87-4b77-a4e9-f20ae3af7318"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all prime numbers that do not have a multiple ending in $2015$.\n\nRemember to put your answer on its own line af... |
Two infinite geometric series have the same sum. The first term of the first series is $1$, and the first term of the second series is $4$. The fifth terms of the two series are equal. The sum of each series can be written as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$. | 25 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTwo infinite geometric series have the same sum. The first term of the first series is $1$, and the first term of the ... | MATH | {
"ground_truth": "25",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "267269e0-dab0-4f95-be12-845dc4ef7d76"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTwo infinite geometric series have the same sum. The first term of the first series is $1$, and the first term of the ... |
For any integer $n \ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ whenever $m$ divides the digit sum of $N$ written in base $b_n$, for $2 \le m \le n$. Find the integer nearest to $b_{36}/b_{25}$. | 5394 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor any integer $n \\ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ wh... | MATH | {
"ground_truth": "5394",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2a7ce3d6-bd08-452f-a52c-115c9a987a36"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor any integer $n \\ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ wh... |
What is the greatest common divisor of $121^2 + 233^2 + 345^2$ and $120^2 + 232^2 + 346^2$? | 5 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the greatest common divisor of $121^2 + 233^2 + 345^2$ and $120^2 + 232^2 + 346^2$?\n\nRemember to put your an... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "56141ea4-7fab-4ab1-961a-0129fcc2fb4d"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the greatest common divisor of $121^2 + 233^2 + 345^2$ and $120^2 + 232^2 + 346^2$?\n\nRemember to put your an... |
The sequence $a_1, a_2, \cdots$ is geometric with $a_1=a$ and common ratio $r,$ where $a$ and $r$ are positive integers. Given that $\log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r).$ | 46 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe sequence $a_1, a_2, \\cdots$ is geometric with $a_1=a$ and common ratio $r,$ where $a$ and $r$ are positive intege... | MATH | {
"ground_truth": "46",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e59a2d40-5cb6-4961-8e54-c7e9cf7c6d8d"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe sequence $a_1, a_2, \\cdots$ is geometric with $a_1=a$ and common ratio $r,$ where $a$ and $r$ are positive intege... |
How many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring of length at least two?
(Note: A substring is a continuous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.) | 6 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8d8602ca-269c-4646-a572-2c5638b43245"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring... |
How many distinct, non-equilateral triangles with a perimeter of 60 units have integer side lengths $a$, $b$, and $c$ such that $a$, $b$, $c$ is an arithmetic sequence? | 9 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many distinct, non-equilateral triangles with a perimeter of 60 units have integer side lengths $a$, $b$, and $c$ ... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e359bce5-be8b-495d-9df8-ab01be8ad99f"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many distinct, non-equilateral triangles with a perimeter of 60 units have integer side lengths $a$, $b$, and $c$ ... |
Let $f(n)$ be the integer closest to $\sqrt[4]{n}.$ Find $\sum\limits_{k=1}^{1995}\frac 1{f(k)}.$ | 400 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f(n)$ be the integer closest to $\\sqrt[4]{n}.$ Find $\\sum\\limits_{k=1}^{1995}\\frac 1{f(k)}.$\n\nRemember to p... | MATH | {
"ground_truth": "400",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c325f43a-9270-4e39-86f7-6338b5cd3d6f"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f(n)$ be the integer closest to $\\sqrt[4]{n}.$ Find $\\sum\\limits_{k=1}^{1995}\\frac 1{f(k)}.$\n\nRemember to p... |
We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24+k$? | 8 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWe know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ suc... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "357043c7-4bff-4b2e-b1a9-f2658989b953"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWe know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ suc... |
The function $f$ is defined by
$$f(x) = \lfloor|x|\rfloor - |\lfloor x \rfloor|$$
for all real numbers $x$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$. The original answer is a set of distinct integer values that form the range of $f$, in the form \{a, b\}. Please... | -1 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe function $f$ is defined by \n$$f(x) = \\lfloor|x|\\rfloor - |\\lfloor x \\rfloor|$$ \nfor all real numbers $x$, wh... | MATH | {
"ground_truth": "-1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c169061c-1a03-4d0d-acfb-eaff26e11035"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe function $f$ is defined by \n$$f(x) = \\lfloor|x|\\rfloor - |\\lfloor x \\rfloor|$$ \nfor all real numbers $x$, wh... |
Two points $A$ and $B$ are randomly chosen on a circle with a radius of $100$. For a positive integer $x$, denote $P(x)$ as the probability that the length of $AB$ is less than $x$. Find the minimum possible integer value of $x$ such that $P(x) > \frac{2}{3}$. | 174 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTwo points $A$ and $B$ are randomly chosen on a circle with a radius of $100$. For a positive integer $x$, denote $P(x... | MATH | {
"ground_truth": "174",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8153d39f-7925-40c3-be8d-3a7a62d210ca"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTwo points $A$ and $B$ are randomly chosen on a circle with a radius of $100$. For a positive integer $x$, denote $P(x... |
What is the sum of the final three digits of the integer representation of $5^{100}$? | 13 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the sum of the final three digits of the integer representation of $5^{100}$?\n\nRemember to put your answer o... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "14bdd3a4-3bbc-4339-969b-9c69d5cd7239"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the sum of the final three digits of the integer representation of $5^{100}$?\n\nRemember to put your answer o... |
Let $a_1, a_2, a_3, \ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j < k$, $a_j \neq a_k$.
- So $1 \in S$;
- $2 \in S$ if and o... | 4030 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a_1, a_2, a_3, \\ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a rand... | MATH | {
"ground_truth": "4030",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "833a8665-dd62-4e41-a94b-f04ffe7da25b"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a_1, a_2, a_3, \\ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a rand... |
Let $\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\Gamma$ with diameter $\overline{BC}$ is constructed outside the triangle. Suppose there exists a circle $\omega$ tangent to $AB$ and $AC$ and furthermore internally tangent to $\Gamma$ at a point $X$. The length $AX$ can be written in ... | 76 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\\Gamma$ with diameter $\\overlin... | MATH | {
"ground_truth": "76",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "de73c551-1c3d-4204-a09a-cbcac6651579"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\\Gamma$ with diameter $\\overlin... |
Let $M$ and $m$ be, respectively, the greatest and the least ten-digit numbers that are rearrangements of the digits $0$ through $9$ such that no two adjacent digits are consecutive. Find $M - m$. | 8456173452 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $M$ and $m$ be, respectively, the greatest and the least ten-digit numbers that are rearrangements of the digits $... | MATH | {
"ground_truth": "8456173452",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "591ebdd0-345e-40e5-9798-13059a9e7fe2"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $M$ and $m$ be, respectively, the greatest and the least ten-digit numbers that are rearrangements of the digits $... |
The set $A=\{1,2,3,\cdots, 10\}$ contains the numbers $1$ through $10$. A subset of $A$ of size $n$ is competent if it contains $n$ as an element. A subset of $A$ is minimally competent if it itself is competent, but none of its proper subsets are. Find the total number of minimally competent subsets of $A$. | 55 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe set $A=\\{1,2,3,\\cdots, 10\\}$ contains the numbers $1$ through $10$. A subset of $A$ of size $n$ is competent if... | MATH | {
"ground_truth": "55",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "656e5bea-df6c-47a6-a736-8cf0c788ca0f"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe set $A=\\{1,2,3,\\cdots, 10\\}$ contains the numbers $1$ through $10$. A subset of $A$ of size $n$ is competent if... |
Let $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Friday. The fourth Sunday of year $N+3$ is the $m$th day of January. What is $m$? | 23 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ i... | MATH | {
"ground_truth": "23",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4fd5bbae-ebee-4d55-b574-3d3488663a2c"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ i... |
How many terms of the arithmetic sequence 88, 85, 82, $\dots$ appear before the number $-17$ appears? | 35 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many terms of the arithmetic sequence 88, 85, 82, $\\dots$ appear before the number $-17$ appears?\n\nRemember to ... | MATH | {
"ground_truth": "35",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a51b1b7f-ec48-48fa-af75-08757f579f2a"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many terms of the arithmetic sequence 88, 85, 82, $\\dots$ appear before the number $-17$ appears?\n\nRemember to ... |
On circle $O$, points $C$ and $D$ are on the same side of diameter $\overline{AB}$, $\angle AOC = 30^\circ$, and $\angle DOB = 45^\circ$. Find the ratio of the area of the smaller sector $COD$ to the area of the circle. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 31 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nOn circle $O$, points $C$ and $D$ are on the same side of diameter $\\overline{AB}$, $\\angle AOC = 30^\\circ$, and $\... | MATH | {
"ground_truth": "31",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9a34b46b-bcc5-4373-b547-290b026c96fa"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nOn circle $O$, points $C$ and $D$ are on the same side of diameter $\\overline{AB}$, $\\angle AOC = 30^\\circ$, and $\... |
A unit cube has vertices \(P_1,P_2,P_3,P_4,P_1',P_2',P_3',\) and \(P_4'\). Vertices \(P_2\), \(P_3\), and \(P_4\) are adjacent to \(P_1\), and for \(1\le i\le 4,\) vertices \(P_i\) and \(P_i'\) are opposite to each other. A regular octahedron has one vertex in each of the segments \(P_1P_2\), \(P_1P_3\), \(P_1P_4\), \(... | 9 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA unit cube has vertices \\(P_1,P_2,P_3,P_4,P_1',P_2',P_3',\\) and \\(P_4'\\). Vertices \\(P_2\\), \\(P_3\\), and \\(P... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "28eb6bde-bc87-422e-876c-0f191a565e09"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA unit cube has vertices \\(P_1,P_2,P_3,P_4,P_1',P_2',P_3',\\) and \\(P_4'\\). Vertices \\(P_2\\), \\(P_3\\), and \\(P... |
Vertex \(E\) of equilateral \(\triangle{ABE}\) is in the interior of unit square \(ABCD\). Let \(R\) be the region consisting of all points inside \(ABCD\) and outside \(\triangle{ABE}\) whose distance from \(\overline{AD}\) is between \(\frac{1}{3}\) and \(\frac{2}{3}\). Find the area of \(R\). The original answer is ... | 53 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nVertex \\(E\\) of equilateral \\(\\triangle{ABE}\\) is in the interior of unit square \\(ABCD\\). Let \\(R\\) be the r... | MATH | {
"ground_truth": "53",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "25e8ee95-c11d-4178-bb36-d5ab68c504f5"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nVertex \\(E\\) of equilateral \\(\\triangle{ABE}\\) is in the interior of unit square \\(ABCD\\). Let \\(R\\) be the r... |
13 fractions are formed using each of the numbers $1, 2, \ldots, 26$ exactly once. For example: $\frac{12}{5}, \frac{18}{26}, \ldots$. What is the maximum number of these fractions that can be integers? | 12 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n13 fractions are formed using each of the numbers $1, 2, \\ldots, 26$ exactly once. For example: $\\frac{12}{5}, \\fra... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "97571703-1121-4b40-a5db-69826c4575b7"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n13 fractions are formed using each of the numbers $1, 2, \\ldots, 26$ exactly once. For example: $\\frac{12}{5}, \\fra... |
Define a sequence $\{a_{n}\}_{n=1}^{\infty}$ with $a_{1} = 1$ and $a_{n+1} = a_{n} + \lfloor \sqrt{a_{n}} \rfloor$ for all $n \geq 1$. Determine the smallest $N$ such that $a_{N} > 2017$. | 95 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDefine a sequence $\\{a_{n}\\}_{n=1}^{\\infty}$ with $a_{1} = 1$ and $a_{n+1} = a_{n} + \\lfloor \\sqrt{a_{n}} \\rfloo... | MATH | {
"ground_truth": "95",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cdd711a3-cc37-4803-9999-33a02429acee"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDefine a sequence $\\{a_{n}\\}_{n=1}^{\\infty}$ with $a_{1} = 1$ and $a_{n+1} = a_{n} + \\lfloor \\sqrt{a_{n}} \\rfloo... |
Suppose that $x^{10} + x + 1 = 0$ and $x^{100} = a_0 + a_1x + \ldots + a_9x^9$. Find $a_5$. | 252 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose that $x^{10} + x + 1 = 0$ and $x^{100} = a_0 + a_1x + \\ldots + a_9x^9$. Find $a_5$.\n\nRemember to put your a... | MATH | {
"ground_truth": "252",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e0120ddd-71cd-4d1f-9045-cd921543e571"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose that $x^{10} + x + 1 = 0$ and $x^{100} = a_0 + a_1x + \\ldots + a_9x^9$. Find $a_5$.\n\nRemember to put your a... |
What is the number of terms with rational coefficients among the $1001$ terms of the expression $(x \sqrt[3]{2} + y \sqrt{3})^{1000}$? | 167 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the number of terms with rational coefficients among the $1001$ terms of the expression $(x \\sqrt[3]{2} + y \... | MATH | {
"ground_truth": "167",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ea6fec52-c915-4831-9f9c-7d78440c7d3d"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the number of terms with rational coefficients among the $1001$ terms of the expression $(x \\sqrt[3]{2} + y \... |
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is 20 cents. If she had one more quarter, the average value would be 21 cents. How many dimes does she have in her purse? | 0 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe average value of all the pennies, nickels, dimes, and quarters in Paula's purse is 20 cents. If she had one more ... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "49a68862-d0f3-45b3-a87b-e6e55d3a961e"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe average value of all the pennies, nickels, dimes, and quarters in Paula's purse is 20 cents. If she had one more ... |
Sally's salary in 2006 was $\$37,500$. For 2007, she received a salary increase of $x$ percent. In 2008, she received another salary increase of $x$ percent. In 2009, she experienced a salary decrease of $2x$ percent, resulting in a salary of $\$34,825$.
Suppose instead, Sally had received a $2x$ percent salary decrea... | 34825 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSally's salary in 2006 was $\\$37,500$. For 2007, she received a salary increase of $x$ percent. In 2008, she received... | MATH | {
"ground_truth": "34825",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ebcabb1e-fb25-49fe-aec8-8ea239f71695"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSally's salary in 2006 was $\\$37,500$. For 2007, she received a salary increase of $x$ percent. In 2008, she received... |
For real numbers $x$ and $y$, let $M$ be the maximum value of the expression $x^4y + x^3y + x^2y + xy + xy^2 + xy^3 + xy^4$, subject to $x + y = 3$. Find $\lfloor M \rfloor$. | 36 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor real numbers $x$ and $y$, let $M$ be the maximum value of the expression $x^4y + x^3y + x^2y + xy + xy^2 + xy^3 + ... | MATH | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "737dd441-009d-44d4-a7ec-1855b7cb987a"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor real numbers $x$ and $y$, let $M$ be the maximum value of the expression $x^4y + x^3y + x^2y + xy + xy^2 + xy^3 + ... |
Let the three sides of a triangle be $\ell, m, n$, respectively, satisfying $\ell > m > n$ and $\{\frac{3^\ell}{10^4}\} = \{\frac{3^m}{10^4}\} = \{\frac{3^n}{10^4}\}$, where $\{x\} = x - \lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimeter of such a triangle... | 3003 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet the three sides of a triangle be $\\ell, m, n$, respectively, satisfying $\\ell > m > n$ and $\\{\\frac{3^\\ell}{1... | MATH | {
"ground_truth": "3003",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e349dce2-b127-4656-ac38-9f801508503e"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet the three sides of a triangle be $\\ell, m, n$, respectively, satisfying $\\ell > m > n$ and $\\{\\frac{3^\\ell}{1... |
If $ab \neq 0$ and $|a| \neq |b|$, find the number of distinct values of $x$ satisfying the equation
\[ \frac{x-a}{b} + \frac{x-b}{a} = \frac{b}{x-a} + \frac{a}{x-b} \]. | 3 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf $ab \\neq 0$ and $|a| \\neq |b|$, find the number of distinct values of $x$ satisfying the equation\n\\[ \\frac{x-a... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "96afdf85-dfc8-4d1f-9402-8e777bdcefeb"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf $ab \\neq 0$ and $|a| \\neq |b|$, find the number of distinct values of $x$ satisfying the equation\n\\[ \\frac{x-a... |
In triangle $ABC$, $\tan \angle CAB = \frac{22}{7}$, and the altitude from $A$ divides $\overline{BC}$ into segments of length $3$ and $17$. What is the area of triangle $ABC$? | 110 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn triangle $ABC$, $\\tan \\angle CAB = \\frac{22}{7}$, and the altitude from $A$ divides $\\overline{BC}$ into segmen... | MATH | {
"ground_truth": "110",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0eb28a6a-b591-4cb4-a4c3-abf4e9f96759"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn triangle $ABC$, $\\tan \\angle CAB = \\frac{22}{7}$, and the altitude from $A$ divides $\\overline{BC}$ into segmen... |
The sum \(\sum_{k=3}^{\infty} \frac{1}{k(k^4-5k^2+4)^2}\) is equal to \(\frac{m^2}{2n^2}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\). | 49 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe sum \\(\\sum_{k=3}^{\\infty} \\frac{1}{k(k^4-5k^2+4)^2}\\) is equal to \\(\\frac{m^2}{2n^2}\\), where \\(m\\) and ... | MATH | {
"ground_truth": "49",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "86fe139a-a621-45aa-b7ec-66a45c5431aa"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe sum \\(\\sum_{k=3}^{\\infty} \\frac{1}{k(k^4-5k^2+4)^2}\\) is equal to \\(\\frac{m^2}{2n^2}\\), where \\(m\\) and ... |
Find the smallest value of the expression $$(x-1)(x-2)(x-3)(x-4) + 10$$. | 9 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the smallest value of the expression $$(x-1)(x-2)(x-3)(x-4) + 10$$.\n\nRemember to put your answer on its own lin... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "33b523e9-e20c-4129-b837-aafd1cd28e9a"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the smallest value of the expression $$(x-1)(x-2)(x-3)(x-4) + 10$$.\n\nRemember to put your answer on its own lin... |
There are $n$ sticks which have distinct integer lengths. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What is the largest possible value of $n$ under such conditions? | 6 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are $n$ sticks which have distinct integer lengths. Suppose that it's possible to form a non-degenerate triangle... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d49ee66b-4483-40ff-8f63-c559dc8735b4"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are $n$ sticks which have distinct integer lengths. Suppose that it's possible to form a non-degenerate triangle... |
A geometric progression of positive integers has $n$ terms; the first term is $10^{2015}$ and the last term is an odd positive integer. How many possible values of $n$ are there? | 8 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA geometric progression of positive integers has $n$ terms; the first term is $10^{2015}$ and the last term is an odd ... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e6b321b4-8712-41ba-b07b-3d88c7189ecf"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA geometric progression of positive integers has $n$ terms; the first term is $10^{2015}$ and the last term is an odd ... |
Let $f(n)$ and $g(n)$ be functions satisfying
$$f(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}$$
and
$$g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}$$
for positive integer... | 258 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f(n)$ and $g(n)$ be functions satisfying \n$$f(n) = \\begin{cases}\\sqrt{n} & \\text{ if } \\sqrt{n} \\text{ is ... | MATH | {
"ground_truth": "258",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6ba70d09-b680-459f-8b84-5f9cc94d4d2c"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f(n)$ and $g(n)$ be functions satisfying \n$$f(n) = \\begin{cases}\\sqrt{n} & \\text{ if } \\sqrt{n} \\text{ is ... |
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38$, $38$, and $34$. What is the distance between two adjacent parallel lines? | 6 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThree equally spaced parallel lines intersect a circle, creating three chords of lengths $38$, $38$, and $34$. What is... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7f50f4f9-6faf-4766-a83b-4d5cac37c171"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThree equally spaced parallel lines intersect a circle, creating three chords of lengths $38$, $38$, and $34$. What is... |
Let $N$ be the number of functions $f:\{1,2,3,4,5,6,7,8,9,10\} \rightarrow \{1,2,3,4,5\}$ that have the property that for $1\leq x\leq 5$ it is true that $f(f(x))=x$. Given that $N$ can be written in the form $5^a\cdot b$ for positive integers $a$ and $b$ with $b$ not divisible by $5$, find $a+b$. | 31 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $N$ be the number of functions $f:\\{1,2,3,4,5,6,7,8,9,10\\} \\rightarrow \\{1,2,3,4,5\\}$ that have the property ... | MATH | {
"ground_truth": "31",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "09383fff-6443-4c45-aa9c-112c318543f7"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $N$ be the number of functions $f:\\{1,2,3,4,5,6,7,8,9,10\\} \\rightarrow \\{1,2,3,4,5\\}$ that have the property ... |
Loki, Moe, Nick, and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money, and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have? The original answer ... | 5 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLoki, Moe, Nick, and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, ... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "916d3d52-bdcf-4c04-85b5-a10152a9fafa"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLoki, Moe, Nick, and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, ... |
Determine the last (rightmost) three decimal digits of $n$ where:
\[
n=1 \times 3 \times 5 \times 7 \times \ldots \times 2019.
\] | 875 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the last (rightmost) three decimal digits of $n$ where:\n\\[\nn=1 \\times 3 \\times 5 \\times 7 \\times \\ld... | MATH | {
"ground_truth": "875",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1a6182a0-590b-47b2-8604-c40abf0fe0ca"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the last (rightmost) three decimal digits of $n$ where:\n\\[\nn=1 \\times 3 \\times 5 \\times 7 \\times \\ld... |
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$? | 8 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $1... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2eb82fde-325b-4a68-97b5-d9a8b81591ef"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $1... |
Determine the largest possible number of groups one can compose from the integers $1, 2, 3, \ldots, 19, 20$, so that the product of the numbers in each group is a perfect square. Each group may contain exactly one number, in which case the product equals this number, and each number must be in exactly one group. | 8 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the largest possible number of groups one can compose from the integers $1, 2, 3, \\ldots, 19, 20$, so that ... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8bdd157d-1706-4144-b271-5bd1696237c0"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the largest possible number of groups one can compose from the integers $1, 2, 3, \\ldots, 19, 20$, so that ... |
Let $\overline{abc}$ be a three-digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest possible prime factor of $\overline{abc}$? | 29 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\overline{abc}$ be a three-digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest pos... | MATH | {
"ground_truth": "29",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2d397a24-7320-4fe0-a0d9-9d086e071113"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\overline{abc}$ be a three-digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest pos... |
Johny's father tells him: "I am twice as old as you will be seven years from the time I was thrice as old as you were". What is Johny's age? | 14 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nJohny's father tells him: \"I am twice as old as you will be seven years from the time I was thrice as old as you were... | MATH | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a25307ac-464c-4abe-af85-cdb3bae26f5e"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nJohny's father tells him: \"I am twice as old as you will be seven years from the time I was thrice as old as you were... |
Among $100$ points in the plane, with no three collinear, exactly $4026$ pairs are connected by line segments. Each point is then randomly assigned an integer from $1$ to $100$ inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two p... | 1037 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAmong $100$ points in the plane, with no three collinear, exactly $4026$ pairs are connected by line segments. Each po... | MATH | {
"ground_truth": "1037",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3a426642-ab79-45bc-b84c-af0022aee2f8"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAmong $100$ points in the plane, with no three collinear, exactly $4026$ pairs are connected by line segments. Each po... |
Two circles of radius 1 are centered at $(4,0)$ and $(-4,0).$ How many circles are tangent to both of the given circles and also pass through the point $(0,5)$? | 4 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTwo circles of radius 1 are centered at $(4,0)$ and $(-4,0).$ How many circles are tangent to both of the given circle... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e855fcc8-7512-47fc-acc1-2ba3b55af358"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTwo circles of radius 1 are centered at $(4,0)$ and $(-4,0).$ How many circles are tangent to both of the given circle... |
Evaluate the infinite product $$\prod_{n=2}^{\infty}\frac{n^{3}-1}{n^{3}+1}$$.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 5 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEvaluate the infinite product $$\\prod_{n=2}^{\\infty}\\frac{n^{3}-1}{n^{3}+1}$$.The answer is in the form \\frac{m}{n... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "27246872-1256-4162-ad39-2d4422a324e1"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEvaluate the infinite product $$\\prod_{n=2}^{\\infty}\\frac{n^{3}-1}{n^{3}+1}$$.The answer is in the form \\frac{m}{n... |
One rainy afternoon, you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth until you have written the number $99$ ninety-nine times. What is the $2005^{th}$ digit that you write? | 5 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nOne rainy afternoon, you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth unt... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b827d78b-910f-4e7b-9c68-5dd2882d0411"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nOne rainy afternoon, you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth unt... |
Given that for reals $a_1, a_2, \ldots, a_{2004}$, the equation
$$x^{2006} - 2006x^{2005} + a_{2004}x^{2004} + \cdots + a_2x^2 + a_1x + 1 = 0$$
has 2006 positive real solutions, find the maximum possible value of $a_1$. | -2006 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nGiven that for reals $a_1, a_2, \\ldots, a_{2004}$, the equation\n\n$$x^{2006} - 2006x^{2005} + a_{2004}x^{2004} + \\c... | MATH | {
"ground_truth": "-2006",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b59afd2a-c93f-483f-a4dd-65023b84140f"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nGiven that for reals $a_1, a_2, \\ldots, a_{2004}$, the equation\n\n$$x^{2006} - 2006x^{2005} + a_{2004}x^{2004} + \\c... |
How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two? | 112 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two... | MATH | {
"ground_truth": "112",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "71b9c96e-2e7b-4f98-a03a-9bded9ceabbf"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two... |
Sheila is making a regular-hexagon-shaped sign with side length $1$. Let $ABCDEF$ be the regular hexagon, and let $R$, $S$, $T$, and $U$ be the midpoints of $FA$, $BC$, $CD$, and $EF$, respectively. Sheila splits the hexagon into four regions of equal width: trapezoids $ABSR$, $RSCF$, $FCTU$, and $UTDE$. She then paint... | 19 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSheila is making a regular-hexagon-shaped sign with side length $1$. Let $ABCDEF$ be the regular hexagon, and let $R$,... | MATH | {
"ground_truth": "19",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f69437a4-d064-4964-8a20-36ac6d58751f"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSheila is making a regular-hexagon-shaped sign with side length $1$. Let $ABCDEF$ be the regular hexagon, and let $R$,... |
A positive integer $n$ is nice if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$) such that the sum of the four divisors is equal to $n$. Find how many numbers in the set $\{ 2010,2011,2012,\cdots,2019 \}$ are nice. | 1 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA positive integer $n$ is nice if there is a positive integer $m$ with exactly four positive divisors (including $1$ a... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e6dfe8b9-23d4-4ca3-aea6-5f5fca864a09"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA positive integer $n$ is nice if there is a positive integer $m$ with exactly four positive divisors (including $1$ a... |
In the MTRP district, there are $10$ cities. Bob the builder wants to construct roads between these cities such that there is exactly one unique path between any two cities. The government has provided him with a budget of Rs. $20$, and each road requires a positive integer amount (in Rs.) to build. In how many ways ca... | 0 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the MTRP district, there are $10$ cities. Bob the builder wants to construct roads between these cities such that t... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9940e408-0260-4234-a366-673532a8431e"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the MTRP district, there are $10$ cities. Bob the builder wants to construct roads between these cities such that t... |
Harold, Tanya, and Ulysses paint a very long picket fence.
Harold starts with the first picket and paints every $h$th picket;
Tanya starts with the second picket and paints every $t$th picket; and
Ulysses starts with the third picket and paints every $u$th picket.
Call the positive integer $100h+10t+u$ paintable when... | 757 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHarold, Tanya, and Ulysses paint a very long picket fence. \n\nHarold starts with the first picket and paints every $h... | MATH | {
"ground_truth": "757",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fea6ed7d-fd8a-423a-b862-40a710659016"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHarold, Tanya, and Ulysses paint a very long picket fence. \n\nHarold starts with the first picket and paints every $h... |
The $52$ cards in a deck are numbered $1, 2, \cdots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards form a team, and the two persons with higher numbered cards form another team. Let $... | 263 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe $52$ cards in a deck are numbered $1, 2, \\cdots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the de... | MATH | {
"ground_truth": "263",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b0c5638c-f438-48a0-996b-13104825aee1"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe $52$ cards in a deck are numbered $1, 2, \\cdots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the de... |
Determine the smallest positive integer $M$ with the following property: For every choice of integers $a, b, c$, there exists a polynomial $P(x)$ with integer coefficients such that $P(1) = aM$, $P(2) = bM$, and $P(4) = cM$. | 6 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the smallest positive integer $M$ with the following property: For every choice of integers $a, b, c$, there... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d9a9518b-173e-46b4-9af1-30893f39c737"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the smallest positive integer $M$ with the following property: For every choice of integers $a, b, c$, there... |
Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of the number of divisors of $2p^2+2p+1$. | 2 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $p$ be a prime number such that $\\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of the number ... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "10d1802c-65ab-4bd4-a807-4d82d05cd932"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $p$ be a prime number such that $\\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of the number ... |
Let $1 \leq a, b, c, d, e, f, g, h, i \leq 9$ be distinct integers. Denote by $N$ the maximum value of $a \times b \times c$, $d \times e \times f$, $g \times h \times i$. Find the possible minimum value of $N$. | 72 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $1 \\leq a, b, c, d, e, f, g, h, i \\leq 9$ be distinct integers. Denote by $N$ the maximum value of $a \\times b ... | MATH | {
"ground_truth": "72",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "da63f134-1935-4160-acab-b67d45001f88"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $1 \\leq a, b, c, d, e, f, g, h, i \\leq 9$ be distinct integers. Denote by $N$ the maximum value of $a \\times b ... |
Two positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? | 156 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTwo positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a per... | MATH | {
"ground_truth": "156",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "780e5cee-3714-4834-a947-fe6bdfbd5ee9"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTwo positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a per... |
Given the integers $a, b, c, d, e$ are equal to $1, 2, 3, 4, 5$ in some order, with the condition that no two of $a, b, c, d, e$ are equal to the same integer. Additionally, the following conditions are satisfied:
- $b \leq d$
- $c \geq a$
- $a \leq e$
- $b \geq e$
- $d \neq 5$
Determine the value of $a^b + c^d + e$. | 628 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nGiven the integers $a, b, c, d, e$ are equal to $1, 2, 3, 4, 5$ in some order, with the condition that no two of $a, b... | MATH | {
"ground_truth": "628",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0b2fcb82-c861-4ea4-ac5e-dcab30b6e5e3"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nGiven the integers $a, b, c, d, e$ are equal to $1, 2, 3, 4, 5$ in some order, with the condition that no two of $a, b... |
If $w$ is one of the imaginary roots of the equation $x^3=1$, find the value of the product $(1-w+w^2)(1+w-w^2)$. | 4 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf $w$ is one of the imaginary roots of the equation $x^3=1$, find the value of the product $(1-w+w^2)(1+w-w^2)$.\n\nR... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ae88459e-88e9-46d0-b808-a08953305a01"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf $w$ is one of the imaginary roots of the equation $x^3=1$, find the value of the product $(1-w+w^2)(1+w-w^2)$.\n\nR... |
Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color? Express your answer in the form \(\frac{k}{m}\) where the fraction is in its simplest form, and find the value of \(k + m\). | 7 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThree red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that n... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "642d29dd-f812-4d37-bc55-9832b16d7975"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThree red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that n... |
When \(p = \sum_{k=1}^{6} k \ln{k}\), the number \(e^p\) is an integer. What is the largest power of \(2\) that is a factor of \(e^p\)? If the answer is \(2^x\), what is the value of \(x\)? | 16 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhen \\(p = \\sum_{k=1}^{6} k \\ln{k}\\), the number \\(e^p\\) is an integer. What is the largest power of \\(2\\) tha... | MATH | {
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0a97c611-046d-4e9f-a08e-84e900a8708f"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhen \\(p = \\sum_{k=1}^{6} k \\ln{k}\\), the number \\(e^p\\) is an integer. What is the largest power of \\(2\\) tha... |
Find the three-digit number for which the ratio of the number to the sum of its digits is minimal. | 199 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the three-digit number for which the ratio of the number to the sum of its digits is minimal.\n\nRemember to put ... | MATH | {
"ground_truth": "199",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6cf3262b-4faa-41f2-be0c-808460f1ecbd"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the three-digit number for which the ratio of the number to the sum of its digits is minimal.\n\nRemember to put ... |
Which number in the array below is both the largest in its column and the smallest in its row? (Columns go up and down, rows go right and left.)
$$\begin{matrix}10&6&4&3&2\\11&7&14&10&8\\8&3&4&5&9\\13&4&15&12&1\\8&2&5&9&3\\\end{matrix}$$
Fill in the blank with the correct number. | 7 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhich number in the array below is both the largest in its column and the smallest in its row? (Columns go up and down... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ceafda12-f35e-4a97-ae96-359caeb42be6"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhich number in the array below is both the largest in its column and the smallest in its row? (Columns go up and down... |
The $260$ volumes of the *Encyclopedia Galactica* are out of order in the library. Fortunately for the librarian, the books are numbered. Due to his religion, which holds both encyclopedias and the concept of parity in high esteem, the librarian can only rearrange the books two at a time, and then only by switching the... | 130 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe $260$ volumes of the *Encyclopedia Galactica* are out of order in the library. Fortunately for the librarian, the ... | MATH | {
"ground_truth": "130",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a383654e-751f-43b7-a34d-701c5c69c4a6"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe $260$ volumes of the *Encyclopedia Galactica* are out of order in the library. Fortunately for the librarian, the ... |
Determine the smallest number $N$, a multiple of 83, such that $N^2$ has 63 positive divisors. | 1992 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the smallest number $N$, a multiple of 83, such that $N^2$ has 63 positive divisors.\n\nRemember to put your... | MATH | {
"ground_truth": "1992",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2a8ee4cd-df26-4a4b-837d-62c90ab6cd5a"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the smallest number $N$, a multiple of 83, such that $N^2$ has 63 positive divisors.\n\nRemember to put your... |
Steve has three quarters, three nickels and three pennies. If Steve selects three coins at random and without replacement, what is the probability that the total value is exactly 35 cents? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m +... | 31 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSteve has three quarters, three nickels and three pennies. If Steve selects three coins at random and without replacem... | MATH | {
"ground_truth": "31",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "613df862-6d5f-4d06-8556-18726dc55254"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSteve has three quarters, three nickels and three pennies. If Steve selects three coins at random and without replacem... |
A $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece? | 6 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over a... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1a4a707b-a4c8-4b21-a7e5-feabeeea79db"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over a... |
Five congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two neighbors. A sixth circle, congruent to the other five, is placed tangent to two of the five. If this sixth circle is allowed to roll without slipping around the exterior of the figure formed by t... | 720 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFive congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to it... | MATH | {
"ground_truth": "720",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fb0ef667-a1fd-4d1e-b1c1-fcb5fc7dbd44"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFive congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to it... |
A student must choose a program of four courses from a list of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen? | 9 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA student must choose a program of four courses from a list of courses consisting of English, Algebra, Geometry, Histo... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "200f38b1-db08-4414-861a-f510c9da7083"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA student must choose a program of four courses from a list of courses consisting of English, Algebra, Geometry, Histo... |
A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial
$x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2$
are all real can be written in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 37 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of th... | MATH | {
"ground_truth": "37",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "61863c11-8292-4725-9abb-3cee2033950a"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of th... |
There are real numbers $a$ and $b$ such that for every positive number $x$, we have the identity
\[
\tan^{-1} \left( \frac{1}{x} - \frac{x}{8} \right) + \tan^{-1}(ax) + \tan^{-1}(bx)
= \frac{\pi}{2} \, .
\](Throughout this equation, $\tan^{-1}$ means the inverse tangent function, sometimes written $\arctan$.) W... | 7 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are real numbers $a$ and $b$ such that for every positive number $x$, we have the identity\n\\[\n \\tan^{-1} \\... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ada6e0a6-d878-45ce-8a4d-f8be7cae88fa"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are real numbers $a$ and $b$ such that for every positive number $x$, we have the identity\n\\[\n \\tan^{-1} \\... |
Pick out three numbers from $0, 1, \ldots, 9$, such that their sum is an even number and not less than $10$. We have ________ different ways to pick these numbers. | 51 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPick out three numbers from $0, 1, \\ldots, 9$, such that their sum is an even number and not less than $10$. We have ... | MATH | {
"ground_truth": "51",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a41d9b64-379e-4aa6-b586-5b282d3ef1e2"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPick out three numbers from $0, 1, \\ldots, 9$, such that their sum is an even number and not less than $10$. We have ... |
Daniel writes a list of positive integers, each less than or equal to 10, on a board from top to bottom. Next to each number in Daniel's list, Martin writes the number of times that number appears in Daniel's list, creating a new list of the same length. When Martin's list is read from bottom to top, it matches the lis... | 39 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDaniel writes a list of positive integers, each less than or equal to 10, on a board from top to bottom. Next to each ... | MATH | {
"ground_truth": "39",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e6739860-34e4-4c63-8cac-badcfecb98d9"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDaniel writes a list of positive integers, each less than or equal to 10, on a board from top to bottom. Next to each ... |
Calculate the sum of the coordinates of all pairs of positive integers $(n, k)$ such that $k\equiv 0, 3\pmod 4$, $n > k$, and $\displaystyle\sum^n_{i = k + 1} i^3 = (96^2\cdot3 - 1)\displaystyle\left(\sum^k_{i = 1} i\right)^2 + 48^2$ | 103 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCalculate the sum of the coordinates of all pairs of positive integers $(n, k)$ such that $k\\equiv 0, 3\\pmod 4$, $n ... | MATH | {
"ground_truth": "103",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "19b02f3f-fed8-49d8-b4a8-c25ddc9d836c"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCalculate the sum of the coordinates of all pairs of positive integers $(n, k)$ such that $k\\equiv 0, 3\\pmod 4$, $n ... |
Evaluate the sum
$\sum_{k=0}^{\infty}\left(3\cdot\frac{\ln(4k+2)}{4k+2}-\frac{\ln(4k+3)}{4k+3}-\frac{\ln(4k+4)}{4k+4}-\frac{\ln(4k+5)}{4k+5}\right)$
$=3\cdot\frac{\ln2}{2}-\frac{\ln3}{3}-\frac{\ln4}{4}-\frac{\ln5}{5}+3\cdot\frac{\ln6}{6}-\frac{\ln7}{7}-\frac{\ln8}{8}-\frac{\ln9}{9}+3\cdot\frac{\ln10}{10}-\cdots.$
(As u... | 4 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEvaluate the sum\n$\\sum_{k=0}^{\\infty}\\left(3\\cdot\\frac{\\ln(4k+2)}{4k+2}-\\frac{\\ln(4k+3)}{4k+3}-\\frac{\\ln(4k... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "909797a2-3f29-413b-bce1-8977e3703de9"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEvaluate the sum\n$\\sum_{k=0}^{\\infty}\\left(3\\cdot\\frac{\\ln(4k+2)}{4k+2}-\\frac{\\ln(4k+3)}{4k+3}-\\frac{\\ln(4k... |
In the diagram below, we have $AB = 24$ and $\angle ADB =90^\circ$. If $\sin A = \frac23$ and $\sin C = \frac13$, then what is $DC$?
[asy]
pair A,B,C,D;
A = (0,0);
B = (8*sqrt(5),16);
D = (8*sqrt(5),0);
C = (8*sqrt(5) + 32*sqrt(2),0);
draw(D--B--A--C--B);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label... | 34 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the diagram below, we have $AB = 24$ and $\\angle ADB =90^\\circ$. If $\\sin A = \\frac23$ and $\\sin C = \\frac13... | MATH | {
"ground_truth": "34",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "41562573-1d15-4205-8275-09c428a27534"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the diagram below, we have $AB = 24$ and $\\angle ADB =90^\\circ$. If $\\sin A = \\frac23$ and $\\sin C = \\frac13... |
Let $S$ be the set of all subsets of $\{2,3,\ldots,2016\}$ with size $1007$, and for a nonempty set $T$ of numbers, let $f(T)$ be the product of the elements in $T$. Determine the remainder when \[ \sum_{T\in S}\left(f(T)-f(T)^{-1}\right)^2\] is divided by $2017$. Note: For $b$ relatively prime to $2017$, $b^{-1}$ is t... | 2014 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $S$ be the set of all subsets of $\\{2,3,\\ldots,2016\\}$ with size $1007$, and for a nonempty set $T$ of numbers,... | MATH | {
"ground_truth": "2014",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b28abbcc-a06f-47cb-b501-35ce3a14cb7e"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $S$ be the set of all subsets of $\\{2,3,\\ldots,2016\\}$ with size $1007$, and for a nonempty set $T$ of numbers,... |
The numbers $1447$, $1005$ and $1231$ have something in common: each is a $4$-digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there? | 432 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe numbers $1447$, $1005$ and $1231$ have something in common: each is a $4$-digit number beginning with $1$ that has... | MATH | {
"ground_truth": "432",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0f7bbbb4-2462-4a60-ba4a-39b78ca23d1a"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe numbers $1447$, $1005$ and $1231$ have something in common: each is a $4$-digit number beginning with $1$ that has... |
Sequences $a_n$ and $b_n$ are defined for all positive integers $n$ such that $a_1 = 5$, $b_1 = 7$,
$$a_{n+1} = \frac{\sqrt{(a_n+b_n-1)^2+(a_n-b_n+1)^2}}{2},$$
and
$$b_{n+1} = \frac{\sqrt{(a_n+b_n+1)^2+(a_n-b_n-1)^2}}{2}.$$
How many integers $n$ from 1 to 1000 satisfy the property that $a_n, b_n$ form the legs... | 24 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSequences $a_n$ and $b_n$ are defined for all positive integers $n$ such that $a_1 = 5$, $b_1 = 7$, \n\n$$a_{n+1} = \... | MATH | {
"ground_truth": "24",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e791025c-0c68-449d-9414-838f15783b5d"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSequences $a_n$ and $b_n$ are defined for all positive integers $n$ such that $a_1 = 5$, $b_1 = 7$, \n\n$$a_{n+1} = \... |
Let $ABC$ be a right triangle with a right angle at $A$ and side lengths $AC=8$ and $BC=16$. The lines tangent to the circumcircle of $\triangle ABC$ at points $A$ and $B$ intersect at $D$. Let $E$ be the point on side $\overline{AB}$ such that $\overline{AD} \parallel \overline{CE}$. Then $DE^2$ can be written as $\fr... | 451 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABC$ be a right triangle with a right angle at $A$ and side lengths $AC=8$ and $BC=16$. The lines tangent to the ... | MATH | {
"ground_truth": "451",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "22e577e6-3129-43b3-a9bd-d24a7d7528e1"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABC$ be a right triangle with a right angle at $A$ and side lengths $AC=8$ and $BC=16$. The lines tangent to the ... |
How many ways can you mark 8 squares of an $8 \times 8$ chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.) | 21600 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many ways can you mark 8 squares of an $8 \\times 8$ chessboard so that no two marked squares are in the same row ... | MATH | {
"ground_truth": "21600",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "722bd181-6edd-41d1-9b42-c8c7af8d43b6"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many ways can you mark 8 squares of an $8 \\times 8$ chessboard so that no two marked squares are in the same row ... |
How many pairs of integers \((x, y)\) are there such that \(2x + 5y = xy - 1\)? | 4 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many pairs of integers \\((x, y)\\) are there such that \\(2x + 5y = xy - 1\\)?\n\nRemember to put your answer on ... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5e3ea537-345b-4fba-89f5-880cf1a305d9"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many pairs of integers \\((x, y)\\) are there such that \\(2x + 5y = xy - 1\\)?\n\nRemember to put your answer on ... |
Let $p$ be an odd prime and let $\mathbf{Z}_p$ denote (the field of) the integers modulo $p$. How many elements are in the set $$ \left\{x^2: x \in \mathbf{Z}_p\right\} \cap\left\{y^2+1: y \in \mathbf{Z}_p\right\} ? $$ Please provide the number of elements in the intersection when p = 13. | 4 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $p$ be an odd prime and let $\\mathbf{Z}_p$ denote (the field of) the integers modulo $p$. How many elements are i... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2cef9d0d-5549-4ddb-97dc-bdcbed23f82b"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $p$ be an odd prime and let $\\mathbf{Z}_p$ denote (the field of) the integers modulo $p$. How many elements are i... |
The Boston weatherman says there is a 75 percent chance of rain for each day of the Labor Day four-day weekend. If it doesn't rain, then the weather will be sunny. Paul and Yuri want it to be sunny one of those days for a World War III preenactment, but if it's sunny for more than one day they won't know what to do wi... | 91 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe Boston weatherman says there is a 75 percent chance of rain for each day of the Labor Day four-day weekend. If it ... | MATH | {
"ground_truth": "91",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "908b14af-bf1b-48c4-a54c-a8a851a6337f"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe Boston weatherman says there is a 75 percent chance of rain for each day of the Labor Day four-day weekend. If it ... |
Find the sum of the first two integers $n > 1$ such that $3^n$ is divisible by $n$ and $3^n - 1$ is divisible by $n - 1$. | 30 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the sum of the first two integers $n > 1$ such that $3^n$ is divisible by $n$ and $3^n - 1$ is divisible by $n - ... | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "765bcfd1-9065-4e32-9197-529bdba801a3"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the sum of the first two integers $n > 1$ such that $3^n$ is divisible by $n$ and $3^n - 1$ is divisible by $n - ... |
The sum of the two $5$-digit numbers $AMC10$ and $AMC12$ is $123422$. What is $A+M+C$? | 14 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe sum of the two $5$-digit numbers $AMC10$ and $AMC12$ is $123422$. What is $A+M+C$?\n\nRemember to put your answer ... | MATH | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1b30ef91-405e-4582-b0e7-d19469e5ddab"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe sum of the two $5$-digit numbers $AMC10$ and $AMC12$ is $123422$. What is $A+M+C$?\n\nRemember to put your answer ... |
Let $a, b$, and $c$ be real numbers such that $3^a = 125$, $5^b = 49$,and $7^c = 8$1.
Find the product $abc$. | 24 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a, b$, and $c$ be real numbers such that $3^a = 125$, $5^b = 49$,and $7^c = 8$1.\nFind the product $abc$.\n\nRem... | MATH | {
"ground_truth": "24",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a1576f24-d84a-4d9a-b112-3e2118039bbe"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a, b$, and $c$ be real numbers such that $3^a = 125$, $5^b = 49$,and $7^c = 8$1.\nFind the product $abc$.\n\nRem... |
An equiangular hexagon has side lengths $1, 1, a, 1, 1, a$ in that order. Given that there exists a circle that intersects the hexagon at $12$ distinct points, we have $M < a < N$ for some real numbers $M$ and $N$. Determine the minimum possible value of the ratio $\frac{N}{M}$. | 2 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAn equiangular hexagon has side lengths $1, 1, a, 1, 1, a$ in that order. Given that there exists a circle that inters... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3b6455ff-79fc-4221-adb1-ae9f78fc4ccd"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAn equiangular hexagon has side lengths $1, 1, a, 1, 1, a$ in that order. Given that there exists a circle that inters... |
In the complex plane, the graph of $|z - 3| = 2|z + 3|$ intersects the graph of $|z| = k$ in exactly one point. Find all possible values of $k.$
Enter all possible values, separated by commas. | 9 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the complex plane, the graph of $|z - 3| = 2|z + 3|$ intersects the graph of $|z| = k$ in exactly one point. Find ... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fe57099a-df8b-4ab7-b444-584420d81afc"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the complex plane, the graph of $|z - 3| = 2|z + 3|$ intersects the graph of $|z| = k$ in exactly one point. Find ... |
What are the last two digits in the sum of factorials of the first 100 positive integers? | 13 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat are the last two digits in the sum of factorials of the first 100 positive integers?\n\nRemember to put your answ... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3d80174c-0e1b-4387-9776-af0a95569e9d"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat are the last two digits in the sum of factorials of the first 100 positive integers?\n\nRemember to put your answ... |
Let $(a_1,a_2,a_3,\ldots,a_{12})$ be a permutation of $(1,2,3,\ldots,12)$ for which
$a_1>a_2>a_3>a_4>a_5>a_6 \mathrm{\ and \ } a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}.$
An example of such a permutation is $(6,5,4,3,2,1,7,8,9,10,11,12).$ Find the number of such permutations. | 462 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $(a_1,a_2,a_3,\\ldots,a_{12})$ be a permutation of $(1,2,3,\\ldots,12)$ for which\n$a_1>a_2>a_3>a_4>a_5>a_6 \\math... | MATH | {
"ground_truth": "462",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a39e8700-2750-4c4d-b07d-14ff6aca429b"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $(a_1,a_2,a_3,\\ldots,a_{12})$ be a permutation of $(1,2,3,\\ldots,12)$ for which\n$a_1>a_2>a_3>a_4>a_5>a_6 \\math... |
In the diagram, what is the value of $y$?
[asy]
draw((0,0)--(18,0),black+linewidth(1));
draw((18,0)--(18,-6),black+linewidth(1));
draw((0,0)--(4,6)--(18,-6),black+linewidth(1));
draw((18,0)--(18,-0.5)--(17.5,-0.5)--(17.5,0)--cycle,black+linewidth(1));
label("$80^{\circ}$",(4.5,5),S);
label("$60^{\circ}$",(1,0),NE);
la... | 50 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the diagram, what is the value of $y$?\n\n[asy]\ndraw((0,0)--(18,0),black+linewidth(1));\ndraw((18,0)--(18,-6),blac... | MATH | {
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "20a59def-c61c-4800-9057-734e15284bbe"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the diagram, what is the value of $y$?\n\n[asy]\ndraw((0,0)--(18,0),black+linewidth(1));\ndraw((18,0)--(18,-6),blac... |
Let $\theta$ be an acute angle such that
\[\sin 5 \theta = \sin^5 \theta.\]Compute $\tan 2 \theta.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 4 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\theta$ be an acute angle such that\n\\[\\sin 5 \\theta = \\sin^5 \\theta.\\]Compute $\\tan 2 \\theta.$The answe... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7f4354af-cad7-48ee-b016-8cbd61f6a1c5"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\theta$ be an acute angle such that\n\\[\\sin 5 \\theta = \\sin^5 \\theta.\\]Compute $\\tan 2 \\theta.$The answe... |
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